Definition df-oexpi 6007

Description: Define the ordinal exponentiation operation.

This definition is similar to a conventional definition of exponentiation except that it defines ∅ ↑_𝑜 𝐴 to be 1_𝑜 for all 𝐴 ∈ On, in order to avoid having different cases for whether the base is ∅ or not. (Contributed by Mario Carneiro, 4-Jul-2019.)

Assertion

Ref	Expression
df-oexpi	⊢ ↑𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦))

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-oexpi

Step	Hyp	Ref	Expression
1		coei 6000	. 2 class ↑𝑜
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		con0 4100	. . 3 class On
5	3	cv 1242	. . . 4 class 𝑦
6		vz	. . . . . 6 setvar 𝑧
7		cvv 2557	. . . . . 6 class V
8	6	cv 1242	. . . . . . 7 class 𝑧
9	2	cv 1242	. . . . . . 7 class 𝑥
10		comu 5999	. . . . . . 7 class ·𝑜
11	8, 9, 10	co 5512	. . . . . 6 class (𝑧 ·𝑜 𝑥)
12	6, 7, 11	cmpt 3818	. . . . 5 class (𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥))
13		c1o 5994	. . . . 5 class 1𝑜
14	12, 13	crdg 5956	. . . 4 class rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)
15	5, 14	cfv 4902	. . 3 class (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)
16	2, 3, 4, 4, 15	cmpt2 5514	. 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦))
17	1, 16	wceq 1243	1 wff ↑𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦))

This definition is referenced by:  fnoei  6032  oeiexg  6033  oeiv  6036